Meanwhile, in a large skillet sauté onions and garlic in olive oil for 2-3 minutes over medium heat. Add thyme and roasted peppers, sauté for 1 minute.
Cook pasta according to package directions, adding the broccoli during the last 4 minutes.
Reserve ½ cup of the pasta-cooking water then add it to the sauce, season to taste.
Drain pasta/broccoli and add to the sauce, toss to combine. Finish with cheese before serving.
